Siddhartha Mishra

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We design arbitrarily high-order accurate entropy stable schemes for systems of conservation laws. The schemes, termed TeCNO schemes, are based on two main ingredients: (i) high-order accurate entropy conservative fluxes and (ii) suitable numerical diffusion operators involving ENO reconstructed cell-interface values of scaled entropy variables. Numerical(More)
We consider scalar hyperbolic conservation laws in several (d ≥ 1) spatial dimensions with stochastic initial data. We prove existence and uniqueness of a random-entropy solution and show existence of statistical moments of any order k of this random entropy solution. We present a class of numerical schemes of multi-level Monte Carlo Finite Volume(More)
We consider non-strictly hyperbolic systems of conservation laws in triangular form, which turn up in applications like three-phase flows in porous media flow. We device finite volume schemes of Godunov type for these systems that exploit the triangular structure. We prove that the finite volume schemes converge to weak solutions as the discretization(More)
We extend the Multi-Level Monte Carlo (MLMC) algorithm of [19] in order to quantify uncertainty in the solutions of multi-dimensional hyperbolic systems of conservation laws with uncertain initial data. The algorithm is presented and several issues arising in the massively parallel numerical implementation are addressed. In particular, we present a novel(More)
We consider a scalar conservation law with a discontinuous flux function. The fluxes are non-convex, have multiple points of extrema and can have arbitrary intersections. We propose an entropy formulation based on interface connections and associated jump conditions at the interface. We show that the entropy solutions with respect to each choice of(More)
The replacement of teeth by implants is usually restricted to patients with completed craniofacial growth. The aim of this literature review is to discuss the use of dental implants in normal growing patients and in patients with ectodermal dysplasia and the influence of maxillary and mandibular skeletal and dental growth on the stability of those implants.(More)
We introduce a class of numerical schemes that preserve a discrete version of vorticity in conservation laws which involve grad advection. These schemes are based on reformulating finite volume schemes in terms of vertex centered numerical potentials. The resulting potential-based schemes have a genuinely multidimensional structure. A suitable choice of(More)
We present a streamline diffusion shock capturing spacetime discontinuous Galerkin (DG) method to approximate nonlinear systems of conservation laws in several space dimensions. The degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes. We show entropy stability of the(More)
The Multi-Level Monte Carlo finite volumes (MLMC-FVM) algorithm was shown to be a robust and fast solver for uncertainty quantification in the solutions of multi-dimensional systems of stochastic conservation laws. A novel load balancing procedure is used to ensure scalability of the MLMC algorithm on massively parallel hardware. We describe this procedure(More)
We propose an alternative framework for designing genuinely multi-dimensional (GMD) finite volume schemes for systems of conservation laws in two space dimensions. The approach is based on reformulating edge centered numerical fluxes in terms of vertex centered potentials. Any consistent numerical flux can be used to define the potentials. Suitable choices(More)